Abstract

Narrow bandgap semiconductors exhibit very large optical nonlinearities in the infrared owing to large two-photon absorption that scales as the inverse cube of the bandgap energy and the large losses and refraction from two-photon generated free carriers. Except for extremely short pulses, the free-carrier effects dominate the nonlinear losses and nonlinear refraction. Here we develop a method for the calculation of the free-electron refraction cross section in InSb. We also calculate the Auger recombination coefficient in InSb and find it to be in good agreement with existing experimental data. In all the calculations we rely on Fermi–Dirac statistics and use a four-band k⋅p theory for band structure calculations. Experiments on the transmission of submicrosecond CO2 laser pulses through InSb produce results consistent with the calculated parameters.

© 2008 Optical Society of America

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References

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  1. D. A. Miller, C. T. Seaton, M. E. Prise, and S. D. Smith, "Band-gap-resonant nonlinear refraction in III-V semiconductors," Phys. Rev. Lett. 47, 197-200 (1981).
    [CrossRef]
  2. E. W. Van Stryland, M. A. Woodall, H. Vanherzeele, and M. J. Soileau, "Energy band-gap dependence of two-photon absorption," Opt. Lett. 10, 490-492 (1985).
    [CrossRef] [PubMed]
  3. B. S. Wherrett, "Scaling rules for multiphoton interband absorption in semiconductors," J. Opt. Soc. Am. B 1, 67-72 (1984).
    [CrossRef]
  4. E. W. Van Stryland, H. Vanherzeele, M. A. Woodall, M. J. Soileau, A. L. Smirl, S. Guha, and T. F. Boggess, "Two photon absorption, nonlinear refraction, and optical limiting in semiconductors," Opt. Eng. (Bellingham) 24, 613-623 (1985).
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    [CrossRef]
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    [CrossRef]
  7. E. Van Stryland and L. Chase, "Two photon absorption: inorganic materials," in Handbook of Laser Science and Technology; Supplement 2: Optical Materials, Sec. 8, M.Weber, ed. (CRC Press, 1994), pp. 299-328.
  8. S. W. Kurnick and J. M. Powell, "Optical absorption in pure single crystal InSb at 298 and 78 K," Phys. Rev. 116, 597-604 (1959).
    [CrossRef]
  9. M. P. Hasselbeck, E. W. Van Stryland, and M. Sheik-Bahae, "Dynamic band unblocking and leakage two-photon absorption in InSb," Phys. Rev. B 56, 7395-7403 (1997).
    [CrossRef]
  10. V. Chazapis, H. A. Blom, K. L. Vodopyanov, A. G. Norman, and C. C. Phillips, "Midinfrared picosecond spectroscopy studies of Auger recombination in InSb," Phys. Rev. B 52, 2516-2521 (1995).
    [CrossRef]
  11. P. T. Landsberg, Recombination in Semiconductors (Cambridge U. Press, 1991).
  12. A. R. Beattie, "Auger transitions in semiconductors and their computation," J. Phys. C 18, 6501-6515 (1985).
    [CrossRef]
  13. P. T. Landsberg and A. R. Beattie, "Auger effect in semiconductors," J. Phys. Chem. Solids 8, 73-75 (1959).
    [CrossRef]
  14. C. Kittel, Introduction to Solid State Physics, 7th ed. (Wiley, 1996).
  15. A. Haug, "Carrier density dependence of Auger recombination," Solid-State Electron. 21, 1281-1284 (1978).
    [CrossRef]
  16. M. Combescot and R. Combescot, "Auger recombination in direct-gap semiconductors: effect of anisotropy and warping," Phys. Rev. B 37, 8781-8790 (1988).
    [CrossRef]
  17. M. G. Burt, S. Brand, C. Smith, and R. A. Abram, "Overlap integrals for Auger recombination in direct-bandgap semiconductors: calculation for conduction and havy-hole bands in GaAs and InP," J. Phys. C 17, 6385-6401 (1984).
    [CrossRef]
  18. M. Takeshima, "Auger recombination in InAs, GaSb, InP, and GaAs," J. Appl. Phys. 43, 4114-4119 (1972).
    [CrossRef]
  19. E. O. Kane, "Band structure of indium antimonide," J. Phys. Chem. Solids 1, 249-261 (1957).
    [CrossRef]
  20. M. Cardona and F. H. Pollak, "Energy-band structure of germanium and silicon: the k⋅p method," Phys. Rev. 142, 530-543 (1966).
    [CrossRef]
  21. J. R. Chelikowski and M. L. Cohen, "Nonlocal pseudopotential calculations of the electronic structure of eleven diamond and zinc-blend semiconductors," Phys. Rev. B 14, 556-582 (1976).
    [CrossRef]
  22. P. Scharoch and R. A. Abram, "A method for determining the overlap integrals used in calculations of Auger transition rates in semiconductors," Semicond. Sci. Technol. 3, 973-978 (1988).
    [CrossRef]
  23. P. O. Lowdin, "A note on the quantum-mechanical perturbation theory," J. Chem. Phys. 19, 1396-1401 (1951).
    [CrossRef]
  24. A. R. Beattie, R. A. Abram, and P. Scharoch, "Realistic evaluation of impact ionisation and Auger recombination rates for ccch transition in InSb and InGaAsP," Semicond. Sci. Technol. 5, 738-744 (1990).
    [CrossRef]
  25. A. R. Beattie and A. M. White, "An analytic approximation with a wide range of applicability for electron initiated Auger transitions in narrow-gap semiconductors," J. Appl. Phys. 79, 802-813 (1996).
    [CrossRef]
  26. M. E. Flatte, C. H. Grein, T. C. Hasenberg, S. A. Anson, D. J. Jang, J. T. Olesberg, and T. F. Boggess, "Carrier recombination rates in narrow-gap InAs/Ga1−xInxSb-based superlattices," Phys. Rev. B 59, 5745-5750 (1999).
    [CrossRef]
  27. A. Haug, "Auger recombination in direct-gap semiconductors: band-structure effects," J. Phys. C 16, 4159-4172 (1983).
    [CrossRef]
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  29. D. Yevick and W. Bardyszewski, "An introduction to non-equilibrium many-body analyses of optical processes in III-IV semiconductors," in Semiconductors and Semimetals, R.K.Willardson and A.C.Beer, eds. (Academic, 1993), Vol. 39, pp. 318-388.
  30. M. Sheik-bahae, D. J. Hagan, and E. W. Van Stryland, "Dispersion and band-gap scaling of the electronic Kerr effect in solids associated with two-photon absorption," Phys. Rev. Lett. 65, 96-99 (1989).
    [CrossRef]
  31. M. Sheik-Bahae, D. C. Hutchings, D. J. Hagan, and E. W. Van Stryland, "Dispersion of bound electronic nonlinear refraction in solids," IEEE J. Quantum Electron. QE-27, 1296-1309 (1991).
    [CrossRef]
  32. W. Zawadzki, "Electron transport phenomena in small-gap semiconductors," Adv. Phys. 23, 435-522 (1974).
    [CrossRef]
  33. K. Seeger, Semiconductor Physics. An Introduction, 3rd ed. (Springer-Verlag, 1985).
  34. B. S. Werrett, A. C. Walker, and F. A. P. Tooley, Optical Nonlinearities and Instabilities in Semiconductors (Academic, 1988), pp. 239-272.
  35. A. A. Said, M. Sheik-Bahae, D. J. Hagan, T. H. Wei, J. Wang, J. Young, and E. W. Van Stryland, "Determination of bound-electronic and free-carrier nonlinearities in ZnSe, GaAs, CdTe, and ZnTe," J. Opt. Soc. Am. B 9, 405-414 (1992).
    [CrossRef]
  36. E. J. Johnson, "Absorption near the fundamental edge," in Semiconductors and Semimetals, R.K.Willardson and A.C.Beer, eds. (Academic, 1967), Vol. 3, pp. 153-258.
    [CrossRef]
  37. V. Dubikovskiy, "Optical limiting: numerical modeling and experiment," Ph.D. dissertation (University of Central Florida, 2003).
  38. N. V. Tabiryan, B. Ya. Zel'dovich, M. Kreuzer, T. Vogeler, and T. Tschudi, "Higher-dimensionality caustics owing to competing reorientation of a liquid crystal by laser beams," J. Opt. Soc. Am. B 13, 1426-1969 (1996).
    [CrossRef]
  39. D. I. Kovsh, S. Yang, D. J. Hagan, and E. W. Van Stryland, "Nonlinear optical beam propagation for optical limiting," Appl. Opt. 38, 5168-5180 (1999).
    [CrossRef]
  40. F. E. Hernández, S. Yang, E. W. Van Stryland, and D. J. Hagan, "High dynamic range cascaded-focus optical limiter," Opt. Lett. 25, 1180-1182 (2000).
    [CrossRef]
  41. M. Mohebi, P. F. Aiello, G. Reali, M. J. Soileau, and E. W. Van Stryland, "Self-focusing in CS2 at 10.6 μm," Opt. Lett. 10, 396 (1985).
    [CrossRef] [PubMed]
  42. The authors recently became aware of a recent publication by S. Krishnamurthy, Z. G. Yu, L. P. Gonzalez, and S. Guha, "Accurate evaluation of nonlinear absorption coefficients in InAs, InSb, and HgCdTe alloys," J. Appl. Phys. 101, 113104 (2007); these results are consistent with those reported herein.
    [CrossRef]
  43. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical recipes in C. The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, 1992).

2007

The authors recently became aware of a recent publication by S. Krishnamurthy, Z. G. Yu, L. P. Gonzalez, and S. Guha, "Accurate evaluation of nonlinear absorption coefficients in InAs, InSb, and HgCdTe alloys," J. Appl. Phys. 101, 113104 (2007); these results are consistent with those reported herein.
[CrossRef]

2000

1999

D. I. Kovsh, S. Yang, D. J. Hagan, and E. W. Van Stryland, "Nonlinear optical beam propagation for optical limiting," Appl. Opt. 38, 5168-5180 (1999).
[CrossRef]

M. E. Flatte, C. H. Grein, T. C. Hasenberg, S. A. Anson, D. J. Jang, J. T. Olesberg, and T. F. Boggess, "Carrier recombination rates in narrow-gap InAs/Ga1−xInxSb-based superlattices," Phys. Rev. B 59, 5745-5750 (1999).
[CrossRef]

1997

M. P. Hasselbeck, E. W. Van Stryland, and M. Sheik-Bahae, "Dynamic band unblocking and leakage two-photon absorption in InSb," Phys. Rev. B 56, 7395-7403 (1997).
[CrossRef]

1996

A. R. Beattie and A. M. White, "An analytic approximation with a wide range of applicability for electron initiated Auger transitions in narrow-gap semiconductors," J. Appl. Phys. 79, 802-813 (1996).
[CrossRef]

N. V. Tabiryan, B. Ya. Zel'dovich, M. Kreuzer, T. Vogeler, and T. Tschudi, "Higher-dimensionality caustics owing to competing reorientation of a liquid crystal by laser beams," J. Opt. Soc. Am. B 13, 1426-1969 (1996).
[CrossRef]

1995

V. Chazapis, H. A. Blom, K. L. Vodopyanov, A. G. Norman, and C. C. Phillips, "Midinfrared picosecond spectroscopy studies of Auger recombination in InSb," Phys. Rev. B 52, 2516-2521 (1995).
[CrossRef]

1992

1991

M. Sheik-Bahae, D. C. Hutchings, D. J. Hagan, and E. W. Van Stryland, "Dispersion of bound electronic nonlinear refraction in solids," IEEE J. Quantum Electron. QE-27, 1296-1309 (1991).
[CrossRef]

1990

A. R. Beattie, R. A. Abram, and P. Scharoch, "Realistic evaluation of impact ionisation and Auger recombination rates for ccch transition in InSb and InGaAsP," Semicond. Sci. Technol. 5, 738-744 (1990).
[CrossRef]

1989

M. Sheik-bahae, D. J. Hagan, and E. W. Van Stryland, "Dispersion and band-gap scaling of the electronic Kerr effect in solids associated with two-photon absorption," Phys. Rev. Lett. 65, 96-99 (1989).
[CrossRef]

1988

P. Scharoch and R. A. Abram, "A method for determining the overlap integrals used in calculations of Auger transition rates in semiconductors," Semicond. Sci. Technol. 3, 973-978 (1988).
[CrossRef]

M. Combescot and R. Combescot, "Auger recombination in direct-gap semiconductors: effect of anisotropy and warping," Phys. Rev. B 37, 8781-8790 (1988).
[CrossRef]

1987

1986

1985

A. R. Beattie, "Auger transitions in semiconductors and their computation," J. Phys. C 18, 6501-6515 (1985).
[CrossRef]

E. W. Van Stryland, M. A. Woodall, H. Vanherzeele, and M. J. Soileau, "Energy band-gap dependence of two-photon absorption," Opt. Lett. 10, 490-492 (1985).
[CrossRef] [PubMed]

E. W. Van Stryland, H. Vanherzeele, M. A. Woodall, M. J. Soileau, A. L. Smirl, S. Guha, and T. F. Boggess, "Two photon absorption, nonlinear refraction, and optical limiting in semiconductors," Opt. Eng. (Bellingham) 24, 613-623 (1985).

M. Mohebi, P. F. Aiello, G. Reali, M. J. Soileau, and E. W. Van Stryland, "Self-focusing in CS2 at 10.6 μm," Opt. Lett. 10, 396 (1985).
[CrossRef] [PubMed]

1984

B. S. Wherrett, "Scaling rules for multiphoton interband absorption in semiconductors," J. Opt. Soc. Am. B 1, 67-72 (1984).
[CrossRef]

M. G. Burt, S. Brand, C. Smith, and R. A. Abram, "Overlap integrals for Auger recombination in direct-bandgap semiconductors: calculation for conduction and havy-hole bands in GaAs and InP," J. Phys. C 17, 6385-6401 (1984).
[CrossRef]

1983

A. Haug, "Auger recombination in direct-gap semiconductors: band-structure effects," J. Phys. C 16, 4159-4172 (1983).
[CrossRef]

1981

D. A. Miller, C. T. Seaton, M. E. Prise, and S. D. Smith, "Band-gap-resonant nonlinear refraction in III-V semiconductors," Phys. Rev. Lett. 47, 197-200 (1981).
[CrossRef]

1980

L. A. Almazov, A. I. Liptuga, V. K. Malyutenko, and L. L. Fedorenko, Fiz. Tekh. Poluprovodn. (S.-Peterburg) 14, 1940 (1980) L. A. Almazov, A. I. Liptuga, V. K. Malyutenko, and L. L. Fedorenko,[Sov. Phys. Semicond. 14, 1154 (1980)].

1978

A. Haug, "Carrier density dependence of Auger recombination," Solid-State Electron. 21, 1281-1284 (1978).
[CrossRef]

1976

J. R. Chelikowski and M. L. Cohen, "Nonlocal pseudopotential calculations of the electronic structure of eleven diamond and zinc-blend semiconductors," Phys. Rev. B 14, 556-582 (1976).
[CrossRef]

1974

W. Zawadzki, "Electron transport phenomena in small-gap semiconductors," Adv. Phys. 23, 435-522 (1974).
[CrossRef]

1972

M. Takeshima, "Auger recombination in InAs, GaSb, InP, and GaAs," J. Appl. Phys. 43, 4114-4119 (1972).
[CrossRef]

1966

M. Cardona and F. H. Pollak, "Energy-band structure of germanium and silicon: the k⋅p method," Phys. Rev. 142, 530-543 (1966).
[CrossRef]

1959

S. W. Kurnick and J. M. Powell, "Optical absorption in pure single crystal InSb at 298 and 78 K," Phys. Rev. 116, 597-604 (1959).
[CrossRef]

P. T. Landsberg and A. R. Beattie, "Auger effect in semiconductors," J. Phys. Chem. Solids 8, 73-75 (1959).
[CrossRef]

1957

E. O. Kane, "Band structure of indium antimonide," J. Phys. Chem. Solids 1, 249-261 (1957).
[CrossRef]

1951

P. O. Lowdin, "A note on the quantum-mechanical perturbation theory," J. Chem. Phys. 19, 1396-1401 (1951).
[CrossRef]

Adv. Phys.

W. Zawadzki, "Electron transport phenomena in small-gap semiconductors," Adv. Phys. 23, 435-522 (1974).
[CrossRef]

Appl. Opt.

Fiz. Tekh. Poluprovodn. (S.-Peterburg)

L. A. Almazov, A. I. Liptuga, V. K. Malyutenko, and L. L. Fedorenko, Fiz. Tekh. Poluprovodn. (S.-Peterburg) 14, 1940 (1980) L. A. Almazov, A. I. Liptuga, V. K. Malyutenko, and L. L. Fedorenko,[Sov. Phys. Semicond. 14, 1154 (1980)].

IEEE J. Quantum Electron.

M. Sheik-Bahae, D. C. Hutchings, D. J. Hagan, and E. W. Van Stryland, "Dispersion of bound electronic nonlinear refraction in solids," IEEE J. Quantum Electron. QE-27, 1296-1309 (1991).
[CrossRef]

J. Appl. Phys.

A. R. Beattie and A. M. White, "An analytic approximation with a wide range of applicability for electron initiated Auger transitions in narrow-gap semiconductors," J. Appl. Phys. 79, 802-813 (1996).
[CrossRef]

M. Takeshima, "Auger recombination in InAs, GaSb, InP, and GaAs," J. Appl. Phys. 43, 4114-4119 (1972).
[CrossRef]

The authors recently became aware of a recent publication by S. Krishnamurthy, Z. G. Yu, L. P. Gonzalez, and S. Guha, "Accurate evaluation of nonlinear absorption coefficients in InAs, InSb, and HgCdTe alloys," J. Appl. Phys. 101, 113104 (2007); these results are consistent with those reported herein.
[CrossRef]

J. Chem. Phys.

P. O. Lowdin, "A note on the quantum-mechanical perturbation theory," J. Chem. Phys. 19, 1396-1401 (1951).
[CrossRef]

J. Opt. Soc. Am. B

J. Phys. C

A. R. Beattie, "Auger transitions in semiconductors and their computation," J. Phys. C 18, 6501-6515 (1985).
[CrossRef]

M. G. Burt, S. Brand, C. Smith, and R. A. Abram, "Overlap integrals for Auger recombination in direct-bandgap semiconductors: calculation for conduction and havy-hole bands in GaAs and InP," J. Phys. C 17, 6385-6401 (1984).
[CrossRef]

A. Haug, "Auger recombination in direct-gap semiconductors: band-structure effects," J. Phys. C 16, 4159-4172 (1983).
[CrossRef]

J. Phys. Chem. Solids

P. T. Landsberg and A. R. Beattie, "Auger effect in semiconductors," J. Phys. Chem. Solids 8, 73-75 (1959).
[CrossRef]

E. O. Kane, "Band structure of indium antimonide," J. Phys. Chem. Solids 1, 249-261 (1957).
[CrossRef]

Opt. Eng. (Bellingham)

E. W. Van Stryland, H. Vanherzeele, M. A. Woodall, M. J. Soileau, A. L. Smirl, S. Guha, and T. F. Boggess, "Two photon absorption, nonlinear refraction, and optical limiting in semiconductors," Opt. Eng. (Bellingham) 24, 613-623 (1985).

Opt. Lett.

Phys. Rev.

S. W. Kurnick and J. M. Powell, "Optical absorption in pure single crystal InSb at 298 and 78 K," Phys. Rev. 116, 597-604 (1959).
[CrossRef]

M. Cardona and F. H. Pollak, "Energy-band structure of germanium and silicon: the k⋅p method," Phys. Rev. 142, 530-543 (1966).
[CrossRef]

Phys. Rev. B

J. R. Chelikowski and M. L. Cohen, "Nonlocal pseudopotential calculations of the electronic structure of eleven diamond and zinc-blend semiconductors," Phys. Rev. B 14, 556-582 (1976).
[CrossRef]

M. P. Hasselbeck, E. W. Van Stryland, and M. Sheik-Bahae, "Dynamic band unblocking and leakage two-photon absorption in InSb," Phys. Rev. B 56, 7395-7403 (1997).
[CrossRef]

V. Chazapis, H. A. Blom, K. L. Vodopyanov, A. G. Norman, and C. C. Phillips, "Midinfrared picosecond spectroscopy studies of Auger recombination in InSb," Phys. Rev. B 52, 2516-2521 (1995).
[CrossRef]

M. E. Flatte, C. H. Grein, T. C. Hasenberg, S. A. Anson, D. J. Jang, J. T. Olesberg, and T. F. Boggess, "Carrier recombination rates in narrow-gap InAs/Ga1−xInxSb-based superlattices," Phys. Rev. B 59, 5745-5750 (1999).
[CrossRef]

M. Combescot and R. Combescot, "Auger recombination in direct-gap semiconductors: effect of anisotropy and warping," Phys. Rev. B 37, 8781-8790 (1988).
[CrossRef]

Phys. Rev. Lett.

M. Sheik-bahae, D. J. Hagan, and E. W. Van Stryland, "Dispersion and band-gap scaling of the electronic Kerr effect in solids associated with two-photon absorption," Phys. Rev. Lett. 65, 96-99 (1989).
[CrossRef]

D. A. Miller, C. T. Seaton, M. E. Prise, and S. D. Smith, "Band-gap-resonant nonlinear refraction in III-V semiconductors," Phys. Rev. Lett. 47, 197-200 (1981).
[CrossRef]

Semicond. Sci. Technol.

P. Scharoch and R. A. Abram, "A method for determining the overlap integrals used in calculations of Auger transition rates in semiconductors," Semicond. Sci. Technol. 3, 973-978 (1988).
[CrossRef]

A. R. Beattie, R. A. Abram, and P. Scharoch, "Realistic evaluation of impact ionisation and Auger recombination rates for ccch transition in InSb and InGaAsP," Semicond. Sci. Technol. 5, 738-744 (1990).
[CrossRef]

Solid-State Electron.

A. Haug, "Carrier density dependence of Auger recombination," Solid-State Electron. 21, 1281-1284 (1978).
[CrossRef]

Other

D. Yevick and W. Bardyszewski, "An introduction to non-equilibrium many-body analyses of optical processes in III-IV semiconductors," in Semiconductors and Semimetals, R.K.Willardson and A.C.Beer, eds. (Academic, 1993), Vol. 39, pp. 318-388.

E. J. Johnson, "Absorption near the fundamental edge," in Semiconductors and Semimetals, R.K.Willardson and A.C.Beer, eds. (Academic, 1967), Vol. 3, pp. 153-258.
[CrossRef]

V. Dubikovskiy, "Optical limiting: numerical modeling and experiment," Ph.D. dissertation (University of Central Florida, 2003).

K. Seeger, Semiconductor Physics. An Introduction, 3rd ed. (Springer-Verlag, 1985).

B. S. Werrett, A. C. Walker, and F. A. P. Tooley, Optical Nonlinearities and Instabilities in Semiconductors (Academic, 1988), pp. 239-272.

C. Kittel, Introduction to Solid State Physics, 7th ed. (Wiley, 1996).

E. Van Stryland and L. Chase, "Two photon absorption: inorganic materials," in Handbook of Laser Science and Technology; Supplement 2: Optical Materials, Sec. 8, M.Weber, ed. (CRC Press, 1994), pp. 299-328.

P. T. Landsberg, Recombination in Semiconductors (Cambridge U. Press, 1991).

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical recipes in C. The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, 1992).

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Figures (14)

Fig. 1
Fig. 1

Band structure and transitions in InSb.

Fig. 2
Fig. 2

Experimental and theoretical recombination rates versus excess carrier density. Experimental data is from [10] (plus signs) and [28] (closed triangle). Theoretical rates are calculated based on Eq. (10) for the CCHC Auger process (open squares) and for the CHLH process (open triangles).

Fig. 3
Fig. 3

Comparison of various models for calculation of the Auger recombination rate and experimental data. Detailed band structure calculations from [25] (solid curve), present work calculations with the simplified Kane band structure Eq. (10) without screening (open squares) and with static screening (closed squares), best fit to classical Auger process Eq. (1) (dashed curve), experimental data from [10] (plus signs) and [28] (closed triangle).

Fig. 4
Fig. 4

Refraction cross section versus photoexcited carrier density. Absorption blocking contribution (dot-and-dash curve), free electron refraction contribution (solid curve), total refraction cross section (long dash curve), and 80% of total refraction cross section (short dash curve) producing the best fit to the time resolved measurements shown in Fig 11.

Fig. 5
Fig. 5

Laser pulse temporal profile measured with a fast Au-doped Ge detector.

Fig. 6
Fig. 6

Experimental setup. Squares indicate detectors (D1 reference, D2 sample transmittance), and arrows show distances from the lens to the sample and from the sample to an aperture placed in front of the transmission detector.

Fig. 7
Fig. 7

Spatial beam profile at focus.

Fig. 8
Fig. 8

Beam width measurements (closed triangles) and Gaussian fitting (solid curve) along the propagation path of the focused beam.

Fig. 9
Fig. 9

Beam image obtained at the detector plane D2 with (a) no sample and (b) 1 mm thick InSb sample at the focus.

Fig. 10
Fig. 10

Experimental and numerical results for output versus input energy. Open aperture Z-scan experimental results are shown as open squares and numerical simulation of beam propagation as a long dash curve. For closed aperture Z-scan data the detector is placed on axis 13 mm after the sample. The 5 mm aperture is 6.5 times larger than the beam FW 1 e 2 M at this position in the linear regime. The closed aperture experimental results are shown as closed squares and the numerical simulation of the beam propagation as a short dash curve. Dot-and-dash curve shows numerical simulation results with 70% of the predicted theoretical refraction for comparison.

Fig. 11
Fig. 11

Time-resolved measurements and the best fit with the numerical beam propagation. The refraction cross section σ ref was the only fitting parameter used. The best fit is obtained by using 80% of the predicted theoretical refraction (Fig. 4). Input pulse energy: 58 μ J (experimental—triangles, numerical—solid curve), 125 μ J (experimental—squares, numerical—dashed curve), 200 μ J (experimental—circles, numerical—dot-and-dash curve).

Fig. 12
Fig. 12

Output versus input energy 13 mm after the sample with a 0.3 mm aperture on axis (open triangles) or at the maximum of transmittance (closed triangles if maximum is not on axis). Numerical simulation of beam propagation results are shown as a solid curve. Dashed curve shows numerical simulation results with 70% of the predicted theoretical refraction for comparison.

Fig. 13
Fig. 13

Auger transition rate Ω versus the energy of the final state four for the excess carrier density n = 2.47 × 10 16 cm 3 . The CCHC process results from Eq. (A11) (squares), and the CHLH process results from Eq. (A19) (triangles).

Fig. 14
Fig. 14

Auger transition rate Ω versus the energy of the final state four for the excess carrier density n = 1.72 × 10 18 cm 3 . The CCHC process results from Eq. (A11) (squares), and the CHLH process results from Eq. (A19) (triangles).

Equations (41)

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d ( Δ n ) d t = β 2 I 2 2 ω C Auger Δ n ( n 0 + Δ n ) ( 2 n 0 + Δ n ) ,
M 1234 = e 2 ε ε 0 V s i , s f ( F 14 ( s 1 i , s 1 f ) F 23 ( s 2 i , s 2 f ) λ 2 + ( k 1 k 4 ) 2 F 24 ( s 1 i , s 1 f ) F 13 ( s 2 i , s 2 f ) λ 2 + ( k 2 k 4 ) 2 ) δ k 1 + k 2 , k 3 + k 4 ,
F 13 ( s i , s f ) = V cell 1 u 1 * ( r , s i ) u 3 ( r , s f ) d r ,
λ 2 = e 2 ε ε 0 ( d n d μ c d p h d μ h d p l d μ l ) ,
r = 2 π 1 V k 1 k 2 k 3 k 4 M 1234 2 P E δ ( E 1 + E 2 E 3 E 4 ) ,
P E = f c ( E 1 ) f c ( E 2 ) f h ( E 3 ) ( 1 f c ( E 4 ) ) ( 1 f c ( E 1 ) ) ( 1 f c ( E 2 ) ) ( 1 f h ( E 3 ) ) f c ( E 4 ) ,
f c ( E ) = 1 ( 1 + exp ( E μ c k b T ) ) ,
f h ( E ) = 1 ( 1 + exp ( E μ h k b T ) ) ,
r = 2 π V ( V 8 π 3 ) 3 M 1234 2 P E δ ( E ) δ ( k 1 + k 2 k 3 k 4 ) d 3 k 1 d 3 k 2 d 3 k 3 d 3 k 4 ,
r = 4 π e 4 256 π 8 ε 2 ε 0 2 k 4 th k 4 2 d k 4 s i , s f ( F 14 ( s 1 i , s 1 f ) F 23 ( s 2 i , s 2 f ) λ 2 + ( k 1 k 4 ) 2 F 24 ( s 1 i , s 1 f ) F 13 ( s 2 i , s 2 f ) λ 2 + ( k 2 k 4 ) 2 ) 2 P E δ ( E ) d 3 k 1 d 3 k 2 ,
σ ref = e 2 λ 4 π ε 0 m c n 0 c 2 ,
m c = m c * ( 1 + 2 E E g ) ,
m c = 0 m c ( E ) N ( E ) f ( E ) d E 0 N ( E ) f ( E ) d E ,
N ( E ) = 1 2 π 2 ( 2 m c * 2 ) 3 2 E ( 1 + E E g ) ( 1 + 2 E E g )
α ( ω ) = 2 m e 2 3 π 2 c n 0 ε 0 ( m c * m ) 3 2 2 m P 2 2 ( ω E g ) E g ω ( 2 ω E g 1 ) ( 1 1 1 + exp [ ( ω E g μ ) k T ] ) ,
Δ n ( ω ) = c π 0 Δ α ( ω ) d ( ω ) ( ω ) 2 ( ω ) 2 ,
Δ α ( x ) = 2 m e 2 k T 3 π 2 c n 0 ε 0 E g 3 2 ( m c * m ) 3 2 2 m P 2 2 x ( 2 x + b ) [ exp ( x g ( n ) ) exp ( x g 0 ) ] x + b [ 1 + exp ( x g 0 ) ] [ 1 + exp ( x g ( n ) ) ] .
Δ n ( a ) = 2 m e 2 3 π 2 c n 0 ε 0 E g 3 2 ( m c * m ) 3 2 2 m P 2 2 0 x ( 2 x + b ) [ exp ( x g ( n ) ) exp ( x g 0 ) ] d x x + b ( x a ) ( x + a + 2 b ) [ 1 + exp ( x g 0 ) ] [ 1 + exp ( x g ( n ) ) ] ,
w = w 0 1 + λ 2 z 2 π 2 w 0 4 ,
E 4 th = E g ( 1 + 2 m c m h ) ,
k 4 th = 2 m c E g ( 1 + 3 m c 2 m h ) .
k 1 = R + m c th m h ( m h + 2 m c th ) S + m c th m h + 2 m c th k 4 ,
k 2 = m h + m c th m h ( m h + 2 m c th ) S + m c th m h + 2 m c th k 4 ,
m c th = 2 k 1 th 2 2 E 1 th ,
R = s ( cos θ 1 , sin θ 1 cos θ 2 , sin θ 1 sin θ 2 cos θ 3 ) ,
S = s sin θ 1 sin θ 2 sin θ 3 ( cos θ 4 , sin θ 4 cos θ 5 , sin θ 4 sin θ 5 ) ,
J ( s , θ i ) = ( m h + m c th m h ( m h + 2 m c th ) ) 3 s 5 sin 4 θ 1 sin 3 θ 2 sin 2 θ 3 sin θ 4 .
s 2 = 2 m c th m h 2 ( m h + m c th ) ( Z ( k 4 ) + W ( k 4 , R , S ) ) ,
Z ( k 4 ) = 3 E g 2 + E g 2 4 + 2 k 4 2 E g 2 m c 2 k 4 2 2 ( m h + 2 m c th ) ,
W ( k 4 , R , S ) = E g E g 2 4 + 2 k 1 2 E g 2 m c E g 2 4 + 2 k 2 2 E g 2 m c + 2 k 1 2 2 m c th + 2 k 2 2 2 m c th ,
r CCHC = 1 2 π 2 k 4 th k 4 2 d k 4 Ω CCHC ( k 4 ) ,
Ω CCHC = e 4 32 π 5 3 ε 2 ε 0 2 m c th ( m h + m c th ) 2 m h ( m h + 2 m c th ) 3 M 2 P s 4 sin 4 θ 1 sin 3 θ 2 sin 2 θ 3 sin θ 4 1 m c th m h 2 ( m h + m c th ) 1 s W s d θ 1 d θ 2 d θ 3 d θ 4 d θ 5 .
E 4 th = E g ( 2 + m c m h ) .
k 4 th = 2 m c E g ( 1 + 3 m c 4 m h ) .
k 1 = R + m h m c th ( 2 m h + m c th ) S + m h 2 m h + m c th k 4
k 2 = m h + m c th m c th ( 2 m h + m c th ) S + m h 2 m h + m c th k 4 ,
m c th = 2 k 3 th 2 2 E 3 th ,
Z ( k 4 ) = 3 E g 2 + E g 2 4 + 2 k 4 2 E g 2 m l 2 k 4 2 2 ( 2 m h + m c th ) ,
W ( k 4 , R , S ) = E g 2 E g 2 4 + 2 k 3 2 E g 2 m c + 2 k 3 2 2 m c th .
r CHLH = 1 2 π 2 k 4 th k 4 2 d k 4 Ω CHLH ( k 4 ) ,
Ω CHLH = e 4 32 π 5 3 ε 2 ε 0 2 m h ( m h + m c th ) 2 m c th ( 2 m h + m c th ) 3 M 2 P s 4 sin 4 θ 1 sin 3 θ 2 sin 2 θ 3 sin θ 4 1 m c th m h 2 ( m h + m c th ) 1 s W s d θ 1 d θ 2 d θ 3 d θ 4 d θ 5 .

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